Stability Analysis of Reaction-Diffusion Equations with Double Diffusivity System

Stability analysis for steady state solution of reaction-diffusion equations with double diffusivity discuss and arise in the solution of problems of flow of homogeneous liquids and heat conduction involving air-temperature ) , ( t x u and a grain-temperature ) t , x ( v , the resulting of this analysis shows that the system is stable when: 4 2 2 2 2 1 2 2 1 1 2 Dk a Lk b Lk b a L b a L 0 ′ ′ ′ ′ ′ ′ + + + − > .

) t , x ( v , the resulting of this analysis shows that the system is stable when:

Introduction
Reaction-diffusion give rise to texture synthesis based on the simulation of a process of local nonlinear interactions, which has been proposed as a model of biological pattern formation. A chemical mechanism that was first proposed by Alan [10] (1952) to account for pattern formation in biological morphogenesis, he postulated that patterning is governed primarily by concurrently operating processes: diffusion of morphogens through the tissue and chemical reactions that produce and destroy morphogens at a rate that depends, among other things, on their concentrations. Such mechanisms are called reaction-diffusion (RD) systems [11]. Antic and Hill (2000) [1] are studied a mathematical model for heat transfer in grain store microclimates, this model is "double diffusivity" such that they are used the heatbalance integral method to transform the coupled partial differential equations to the coupled ordinary differential equations and solved it numerically by using the Fehlberg fourth-fifth order Range-kutta method. Aggarwala and Nasim [2] derived the solution of reaction-diffusion equations with double diffusivity by laplace technique and fourier transforms which appear to be simpler and more direct. Chow Tanya [4] are studied the derivation of similarity solutions for one-dimensional coupled systems of reaction-diffusion equations, these solutions are obtained by means of one-parameter group methods. Molz [7] used a coupled system of a model for water transport through plant tissue and Rubinsein [9] is able to derive a coupled system in a one-dimensional case includes heat conduction in heterogeneous media. In this paper, we will study the stability analysis of steady state solution of double diffusivity model.

Model of equations:
The one-dimensional case of reaction-diffusion equations with double-diffusivity is given by [2] ) t , x ( v denote the air-temperature and a grain-temperature respectively, the self-diffusivities 1 D , 2 D , this system represent mathematical model for heat transfer in grain store microclimates [1]. For dimensionless form, we introduce the following dimensionless quantities: substitute these non-dimensional quantities into equations (1a) , (1b) and the conditions (2a),(2b) we get:

Stability analysis:
Assume that the value of the concentrations u and v has the following from [5]: denote the disturbance case. If we substituted (4) in equations (3a)-(3d), we get the following systems: The steady state system: with the boundary conditions: the second system: with the related boundary conditions:

Solution of the steady state case:
To find the solution to (5a)-(5b) with the boundary conditions (5c)-(5d), we shall use the technique which Benson. D. and sherratt. J. [3] are used for the solution the linearized model of coupled ordinary differential equations as follows: adding (7a) to (7b) gives: we choose (S) such that: which is a quadratic equation for (S) and thus: The general solution of (9) is:

Stability analysis (disturbance case):
Assume that the value of ) t , , has the following form ( [6], [8]): is an eigenvalue represent the speed of the wave, the functions 1 F and 2 F are the constant amplitudes, k is the wave number. The flow is stable if the linearized equation correspond to eigenvalue c with negative part ( ) for presented configurations. Now, substitute (11) in the equations (6a)-(6b), we get respectively: by separate the real part and imaginary part, we get: we substitute (13) in the equation (12b), we get: this system is stable always.
and 0 2  c always, thus the system is stable. or:       , however when the coefficients 2 a and 1 b are increase then the unstable region is increase as shown in figures (1) , (2) and (3) and table (1) and (2).